dx 1₁ (x² + 8x + 13)² press numbers in exact form. Use symbolic notation and fracti - C as much as possible.)
dx 1₁ (x² + 8x + 13)² press numbers in exact form. Use symbolic notation and fracti - C as much as possible.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Evaluate the
∫??(?2+8?+13)2∫dx(x2+8x+13)2
(Express numbers in exact form. Use symbolic notation and fractions where needed. Use ?C for the arbitrary constant. Absorb into ?C as much as possible.)
![---
### Completing the Square Method in Integration
Evaluate the integral by completing the square as a first step:
\[
\int \frac{dx}{(x^2 + 8x + 13)^2}
\]
(Express numbers in exact form. Use symbolic notation and fractions where needed. Use \( C \) for the arbitrary constant. Absorb into \( C \) as much as possible.)
---
The original attempt to solve the integral is displayed below:
\[
\int \frac{dx}{(x^2 + 8x + 13)^2} = \frac{1}{12\sqrt{3}} \ln \left| x + 4 + \sqrt{3} \right| - \frac{1}{12} \frac{1}{(x + 4 + \sqrt{3})} - \frac{1}{12\sqrt{3}} \ln \left| x + 4 - \sqrt{3} \right| - \frac{1}{12} \frac{1}{(x + 4 - \sqrt{3})} + C
\]
(Note: The result is marked as incorrect).
---
### Explanation for Completing the Square
To evaluate the integral correctly, the method of completing the square is utilized to transform the quadratic expression in the denominator into a more integrable form.
1. **Complete the square for \(x^2 + 8x + 13\)**:
\[
x^2 + 8x + 13 = (x^2 + 8x + 16) - 3 = (x + 4)^2 - 3
\]
2. **Rewrite the integral**:
\[
\int \frac{dx}{((x + 4)^2 - 3)^2}
\]
### Continuing the Integration
This process breaks down the more challenging quadratic expression, allowing for easier integration using appropriate substitution methods or known integral forms.
---
For more detailed step-by-step solutions and explanations, please refer to the instructional videos and practice problems provided on our website.
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Transcribed Image Text:---
### Completing the Square Method in Integration
Evaluate the integral by completing the square as a first step:
\[
\int \frac{dx}{(x^2 + 8x + 13)^2}
\]
(Express numbers in exact form. Use symbolic notation and fractions where needed. Use \( C \) for the arbitrary constant. Absorb into \( C \) as much as possible.)
---
The original attempt to solve the integral is displayed below:
\[
\int \frac{dx}{(x^2 + 8x + 13)^2} = \frac{1}{12\sqrt{3}} \ln \left| x + 4 + \sqrt{3} \right| - \frac{1}{12} \frac{1}{(x + 4 + \sqrt{3})} - \frac{1}{12\sqrt{3}} \ln \left| x + 4 - \sqrt{3} \right| - \frac{1}{12} \frac{1}{(x + 4 - \sqrt{3})} + C
\]
(Note: The result is marked as incorrect).
---
### Explanation for Completing the Square
To evaluate the integral correctly, the method of completing the square is utilized to transform the quadratic expression in the denominator into a more integrable form.
1. **Complete the square for \(x^2 + 8x + 13\)**:
\[
x^2 + 8x + 13 = (x^2 + 8x + 16) - 3 = (x + 4)^2 - 3
\]
2. **Rewrite the integral**:
\[
\int \frac{dx}{((x + 4)^2 - 3)^2}
\]
### Continuing the Integration
This process breaks down the more challenging quadratic expression, allowing for easier integration using appropriate substitution methods or known integral forms.
---
For more detailed step-by-step solutions and explanations, please refer to the instructional videos and practice problems provided on our website.
---
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